Like most Brits my age I grew up with a sci-fi Children's program on BBC called Doctor Who. (For those who know, my doctor was Jon Pertwee, although I also saw a good bit of early Tom Baker.) It was actually the longest-running sci-fi TV series in the world, running from 1963-89.
A few years ago it was revived in the UK and has become a big hit - and not just for children. Doctor Who always had themes and scripts that went beyond the children's' audience and the series developed a huge fan base that lived off books and audio series even when the TV series died. The revival follows this with shows that are written to appeal as much to adults as kids. It was really great to sit with a couple my age and their 8 and 10 year old daughters and enjoy the new series. The scripts and acting are good, the only change is that the special effects are also good now (the old special effects made classic star trek look high-tech).
At home we don't watch much telly, the last shows we watched regularly were Buffy and Angel. Cindy, being American, had never seen Doctor Who growing up, but she loves the new series. When we get a new set of DVDs there's usually several nights of "it's late, we're tired, but maybe we can do one more".
If you've never seen Doctor Who the place to start is the opening episode of the revived series: Rose. (Wikipedia has mind-bogglingly comprehensive coverage, but I won't link from here as it's naturally full of spoilers.) Rose not only introduces the set up you'll need for other episodes (who the Doctor is, what the TARDIS is) but also does a really good job of distilling the tone of Doctor Who, capturing the mix of adventure and comedy.
If you like Rose then you can either carry on with the full first season or cherry pick highlights. If you prefer the latter I'll suggest my favorites. My big favorite from the first series was the two part The Empty Child / The Doctor Dances. I rate this as better than most films I've seen, certainly better than most TV. (It won a Hugo award so it's not just me.) It was written by Steven Moffat who is also known for writing the comedy series Coupling. Almost as good is Dalek. It lacks the humor but scores due to a wonderfully intense performance from Christopher Eccleston. I also really like the final two part (Bad Wolf / The Parting of the Ways) but you really need to see the whole series to appreciate it properly. (A tip if you do watch the whole series: don't watch the trailer for Bad Wolf (it appears at the end of Boom Town) as it gives away an important part of the plot of Bad Wolf.)
The second series has a different actor playing the Doctor (they have a nice technobabble rationalization to allow them to change actors easily). It doesn't quite hit the high spot of The Empty Child but is still really good. My suggestions for cherry pickers here would by The Girl in the Fireplace (another Moffat Hugo win) and The Impossible Planet / The Satan Pit.
When I said the second series didn't quite hit the heights of The Empty Child, I'm not being very reasonable because those two episodes are far too good for any TV series to live up to. However the third series (not yet all broadcast in the US - it's good to have friends in the UK) hits those high notes twice. Human Nature / The Family of Blood is a super two-parter that threatens to take away Steven Moffat's crown of writing the best episode. Moffat's response is Blink, which is as good a 45 minutes of TV as you could ever hope for. Not just has it got a great story and some cracking humor, it also achieves Doctor Who's higher purpose. You see Doctor Who is only secondarily about entertainment, it's primary purpose was always to scare the living daylights out of small children. I may be too old now to get behind the sofa, but I do remember how much I enjoyed it.